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The planet is 10 Earth-Radii, with a 1 Earth Gravity value. It is a habitable planet. It has a thin atmosphere. It also has 3 moons and a ring system.
The moons are rather large. The largest one is the size of Earth, and Habitable. The second one is a 1/2 Earth radii, and the smallest one is a 1/4 Earth radii.
Is this system possible? If not, than why, or what can I do to make it possible?
Your planet would have 0.1 x the density of the Earth, i.e. about half the density of water. It might be some kind of weird smallish gas giant with the right chemistry in its atmosphere to support some kind of life, but definitely not a traditional Earthlike habitable planet. It definitely won't fulfil your description of having a thin atmosphere.
Your moons sound okay, given suitable orbital distances that they don't perturb each other's orbits too much. For habitability, the largest habitable moon will require its own magnetic field and a sufficiently thick atmosphere to protect the inhabitants from radiation (magnetic field) and impacts from ring debris (atmosphere). Depending on the composition of your rings there may be frequent (compared to Earth) impacts from ring pieces large enough to reach the ground.
The planet is 10 Earth-Radii, with a 1 Earth Gravity value. It is a habitable planet. It has a thin atmosphere. It also has 3 moons and a ring system.
Apart from the whole "habitable" and "thin atmosphere" thing, you've basically described Saturn. Saturn is ~95x Earth's mass, and 9.5x Earth's radius at its equator. Surface gravity is a bit over 1g, too. Even has a similar number of big moons, and a ring system.
Your requirements match a gas giant just fine, but not a planet with anything approximating an inhabitable surface. Don't be so greedy: you don't need a world that big in order for it to be interesting!
You might be able to have portions of the atmosphere be minimally hostile, with human-compatible temperatures and pressures, but it almost certainly can't be breathable. You can have floating space habitats in such a place where you could go outside on deck in shirtsleeves, so long as you brought breathing gas with you.
The moons are rather large. The largest one is the size of Earth, and Habitable. The second one is a 1/2 Earth radii, and the smallest one is a 1/4 Earth radii.
This isn't totally implausible, but it is stretching plausibility a little. That's a lot of pretty big moons. The one Earth-sized moon is probably just fine. You might have issues with its placement around the parent gas giant, because if Jupiter is anything to go by gas giant's can have very, very strong magnetic fields and associated deadly radiation belts. I'm not sure how best to solve that issue though. I'm not sure if the gas giant has to have a super powerful magnetic field, but it seems likely.
Is this system possible? If not, than why, or what can I do to make it possible?
You have two choices here. One is to be very light on the details, and declare by fiat that it is possible. You'll get away with it just fine.
On the other hand, if you start talking about distances to other worlds, and day and year lengths and all the rest, you start needing to cobble together a working system and that may or may not be practical.
I can recommend hunting down some sort of gravity simulator and fighting with it a bit to see what works. There are quite a lot of complicated factors that fit into these things... n-body spheres of influence and orbital resonances and things, more than I understand or can explain. Start with something simple, see what you can get away with adding.
Orbits are basically falling to Earth and missing. Earth's moon is about 1/4th that of Earth, and while your planet takes up more space than Earth but has the same gravity, it could be possible that an Earth sized moon could occur. The dwarf planet "Charon" is about the same size as Pluto, the body which it orbits, though like manythings about Pluto, it's weird (I rarely hear people refer to Charon/Pluto as co-planets or co-moons though I suspect they might orbit each other... but I can't remember when I learned that, if at all. It's likely from Magic School Bus pc games, if that gives you an idea of how dated it might be).
I'm also not sure if rings are possible, because most moons on the Rocky Planets tend to not be associated with moons... the only one that is sphereoid is Earth's Moon, which was thought to have formed when Earth collided with a planetoid in it's orbit and the impact ejected from both formed the moon. Mars, the only other rocky planet, has Two Moons which are misshapened and believed to be two Asteroids that fell into stable orbit. Mercury and Venus do not have moons. Rocky Planets form after enough asteroids in their orbit collide together to pull in the rest and "clear" their orbit, so I don't think a ring system would form. The Gas giants tend to be more likely to have rings due to the immense size of gas giants (Jupiter is a large reason why there aren't many big rocks from space falling on earth.
The others before me have written plenty about why and how it's not possible normally due to size and density requirements you have set up as a premise. The only way I can think of to make it possible, is make the planet hollow. As long as the outer shell weighs 100x of earth (for 1g, I calculated), the rest should fit into the math more or less. This raises the question of why or how the planet is hollow. Simply, make it an artificial planet. The characters in that world need not be aware of it. Kinda like a lost civilization tech or something. Can be used as a plot point even.
OR you can try filling the center of the planet with something that reduces the gravity, allowing you are 10x planet to have both relevant mass (in order to resemble earth's makeup) and still have 1g at the surface.
